James Cook High School
Decimal Fractions – Teaching Guide
This is a teaching guide that can be used when planning a unit of work on decimal fractions with any ability group. The table below shows some of the common misconceptions demonstrated by students. Students have these misunderstandings because they have little conceptual idea of what a decimal is. To correct this, students will need to develop their own understanding through the use of materials to introduce a concept. If a student does not have any/little understanding of decimals, then it is advisable to focus solely on 1 decimal place.
Questions / Incorrect answers & why?In 56.72 The 5 stands for 5 tens
In 3.58 The 5 stands for 5 ………. / Students read the figure after the point as if it were a whole number, and hence gave the answer of ‘units’
Write in words as you would say 8,030 / ‘eight hundred and thirty – students not seeing the importance of the zero as a place holder.
Add one tenth to 4.254 / 14.254 – confusing tens with tenths.
4.264 – regarding 4.254 as four point two hundred and fifty four
Which is the bigger number 0.45 or 0.8 / 0.45 – regarding 0.45 as zero point forty five.
How many different numbers could you write which lie between 0.41 and 0.42 / Students unable to recognize that a large number of decimals lie between any two numbers.
Multiply 3.6 by ten / 3.60 – students applying ‘add a 0’ rule to decimals!
a) 0.2 x 0.4 = ….. b) 60 ÷ 0.3 = …… / 0.8 – incorrect position of dp.
20 – ignoring the dp, doing 60/3!
2.9 is nearest in size to:
3, 30, 2, 20, 0, 1 / 30 – ignoring dp
2.9 x 7 is nearest in size to:
0.002, 0.02 0.2 2 20 200 / General comment – many students find estimating an answer very difficult
59 ÷ 190 is nearest in size to:
0.003, 0.03, 0.3, 3, 30, 300 / 3 –students swap around the order to do 190/59
Some students say it is not possible to divide by a bigger number.
The cost of a large bag of rice weighing 12.5 kg is $10.95What should be the cost of one kg of rice?
12.5 + 10.95, 10.95 ÷ 12.5, 12.5 ÷ 10.95
10.95 – 12.5, 12.5 – 10.95, 10.95 x 12.5 / Many cannot chose correct operation to use.
Some who knew to divide, put the bigger number first.
Teaching Decimals – A Possible Sequence of Lessons
Lesson Content / Resources1. Introduction to Decimals Fractions(1dp)
Focus on 1 decimal place. Start by introducing to students the need to have decimals. Ask what is one half as a decimal? Many will be able to say 0.5, but how many can explain what the five stands for! Use multicubes to introduce tenths.
Example: Aroha has 6 packets of lollies to share amongst 5 people. How much will they each get? Each person gets 1 whole packet , leaving 1 packet remaining. Now introduce important concept – whenever we have 1 left over, we split it up into 10 equal parts. Splitting 1 whole up into ten parts creates tenths. How many tenths will each person get? So 6 ÷ 5 = 1 whole and 2 tenths. Note deliberate use of words. Also talk about the answer on a calculator – 1.2.
Look at other examples, and model on multicubes and/or diagrams 7 ÷ 2, 8 ÷ 5, 6 ÷ 4, 11 ÷ 2
Decimal Place Value
This activity will help to reinforce students the place of decimals in the number system. Students copy out a place value table(see opposite) and have to breakdown a number / 100 / 10 / 1 /
. E.g 567.4 → 500 + 60 + 7 + / 5 / 6 / 7 / 4
This concept can be extended to hundredths, thousandths etc. The worksheets in the booklets provide extra activities to reinforce the position of tenths. There are some exercises on reading scales. When reading scales, students will make the mistake of only looking at the left number and assume each interval represents 1 unit. See example opposite – here students answer incorrectly 4.1!
A good question to use for a ‘do now’ is to draw a scale with one value on and get the students to put as many answers as they can think of – encourage them to think of decimal and/or fractions. For example on this question; 0 and 100, 40 and 60, 49.5 and 50.5, 49.9 and 50.1 etc / See powerpoints on decimals folder on T drive.
multicubes
decimals booklet
p 1-4
dec/frac/% booklet
p22, 24
2. Adding and Subtracting Decimals(1dp)
There are a variety of materials that can be used to model tenths; multicubes, decimats, pipe numbers. This explanation uses multicubes, but could easily be adapted for use with decimats or pipe numbers. Model on materials or diagram(the powerpoint has good examples) 1.2 + 0.4 – need to make a ‘sleeve’ around 10 multicubes to represent 1.
Pose other similar problems: 1.3 + 0.6, 1.3 + 1.4, .0.9 + 2.2, 2.3 + 3.8
Show on materials subtraction of decimals, e.g 1.2 – 0.7
Pose similar problems: 1.7 – 0.3, 1.3 – 0.6, 2.1 – 0.7, 3.4 – 1.6 ….
There are a number of resources(see opposite column) that can be used to practice add/subtracting decimals. Do not let students use the algorithm. For weaker students, concentrate on just 1dp. / See powerpoints on decimals folder on T drive.
Multicubes
Pipe numbers
decimats
Booklet
p24, 27-33
3. Addition/Subtraction Strategies with Decimals(1dp)
During the addition/subtraction topic(book 5) students will have looked at different strategies to solve addition/subtraction problems. The set of resources in the additional decimals booklet use these methods applied to decimals. Students would benefit first from looking at reading decimal scales(see booklet p31). The example shown opposite is one using the ‘jumping the number line’ strategy.
Strategies covered in ‘additional decimals booklet’:
Compatible decimal fractions / Decimal fractions – tenths / Jumping the number line
Don’t subtract – add! / Problems like 2.3 + ? = 7.1 / Problems like 3.7 + ? = 8.9
Problems like ? + 2.9 = 8.1 / Problems like 7.3 – 1.9 = ? / Near doubles
/ See powerpoints on decimals folder on T drive.
Decimals booklet
p 5-19
4. Rounding Decimals(1dp)
The common rounding rule that “if the digit is over 5, go up” is in fact not always correct.
Successive rounding of each decimal place in turn can lead to errors. For example, to round 4.48 to the nearest whole number, a common incorrect application of the rule is this: Round 4.48 to 4.5, round 4.5 to 5. Yet 4.48 is 4 to the nearest whole number. This issue does not exist when rounding 1dp to the nearest whole number, but it is worth students learning this method now, so that they can apply it to future work beyond 1dp. A reliable method for rounding is this: select the possible rounded number below and the possible rounded number above. See which of these the number to be rounded is closer to. For example, Round 8.6 to the nearest whole number.
Estimating
Move onto using rounding for estimating answers. Students find this a difficult skill to master and they will need to have completed some work on general estimating with whole numbers, before moving onto decimals. To make it easier to begin with, give students questions which are multi choice.
e.g 3.8 x 5 is closet to 2 15 20 200 / See powerpoints on decimals folder on T drive.
Decimals booklet
p 21
5. Multiplying/Dividing Decimals(1dp) by 10, 100, 1000
Students who have learnt the rule ‘add a zero to x by 10’ will incorrectly apply this to decimals e.g 3.4 x 10 = 3.40!!
Consider 3.4 x 10. write 3.4 as 3 wholes and 4 tenths – model using decimats. Now:
10 x 3 wholes = 30 wholes, which gives us 3 tens (30)
10 x 4 tenths = 40 tenths, which gives us 4 wholes, hence 3.4 x 10 = 34.0
Go through similar problems modeling them on the decimats. Students can record the results in a place value table – see below.
100 / 10 / 1 /
3
3 / 4
4
3 / 4
7 / 2
7 / 2
0 / 5
5
2 / 6 / 9
2 / 6 / 9
4 / 0 / 3
4 / 0 / 3
Students can look at these results and try and generalize a method for multiplying by 10 – encourage them to think of the numbers moving, not the decimal point. This concept can then be applied to multiplying by 100 and 1000.
A similar approach can be used for dividing by 10. Using decimats, students can look at how tens, units etc, have to be broken up into ten parts for the division. / See powerpoints on decimals folder on T drive.
Decimats
Decimals booklet
p 20
6. Games/Activities
This is a selection of games/activities that can be used at any stage to help develop students understanding of decimals. They could be used as a ‘do now’ activity, or as a activity within the lesson. They can easily be adapted for work beyond 1dp.
Forward and Backward Number Sequences
Similar to that for whole numbers. Ask students to write down, for example, a number one tenth less than 4.0. What number is one tenth more than 15.9 etc.
PIN
Played in pairs. Students copy out a table with 4 columns – see opposite. One student creates a 4 number PIN (e.g 347.2). The other student has to guess a number in their PIN. The other student tries to guess their PIN. For example, they write 368.4. If the number is in the right spot use a P. If the number is in the students PIN but in the wrong place write a I, if the number is not in their PIN write a N. So for the guess 368.4, we would write PNNI. Students can then swap roles. / 100 / 10 / 1 /
5 / 6 / 7 / 4
Number Hangman
Use the same table as used for the PIN activity. Students play in pairs. One asks the other questions about specific places, like, “is there a five in the tens place?”. They may also ask digit related questions, like, “does the number have the digit eight anywhere?”. Is the tenth digit odd?”, Is the hundreds digit greater than five?”. Each time there is a “no” to a question, a piece is added to the hangman.
ZAP
Students work in pairs. One student begins by putting a number into the calculator, for example, 3.5. Then an instruction is given. “Zap out the 5 in the 3.5”. Here 3.5 – 0.5 “zaps” the 5. Encourage students to use other buttons on the calculator, not just the add or subtract.
Four in a line
See 10 ticks worksheets, level 5 pack 2 page 25
Decimal Archery
See 10 ticks worksheet, level 5 pack 2 page 20.
Possible Sequence of Lessons for Hundredths(and beyond!)
This follows a similar concept as shown for 1dp, however, multicubes cannot be used as they cannot be broken down to show hundredths. Use decimats, or ‘pipe numbers’..
Example: Aroha has 5 bars of chocolate to share amongst 4 people. How much will they each get? Each person gets 1 whole packet , leaving 1 packet remaining. Now follow the golden rule, whenever there is one left over, split it up into ten parts. Splitting 1 whole up into ten parts creates tenths. How many tenths will each person get? Each person has 1 whole and 2 tenths. This leaves 2 tenths, each of which is split into ten parts. Ask students what do we call this smaller part(hundredths). Left with 20 hundredths to share amongst 4 people
So 5 ÷ 4 = 1 whole and 2 tenths and 5 hundredths. Note deliberate use of words. Also talk about the answer on a calculator – 1.25.
Look at other examples, and model on multicubes and/or diagrams
7 ÷ 4, 9 ÷ 4,
Decimal Place Value
This activity will help to reinforce students the place of decimals in the number system. Students copy out a place value table(see opposite) and have to breakdown a number / 100 / 10 / 1 / /
. E.g 507.48 → 500 + 7 + + / 5 / 0 / 7 / 4 / 8
This concept can be extended to thousandths, ten thousandths etc. The worksheets in the booklets provide extra activities to reinforce the position of hundredths. There are some exercises on reading scales. When reading scales, students will make the mistake of only looking at the left number and assume each interval represents 1 unit. See example opposite – here students answer incorrectly 4.1!